DIVERSITY  OF 


N.QN  CIRCULATING 

CHECK  FOR  UNBOUND 
CIRCULATING  COPY 


UNIVERSITY  OF  ILLINOIS 

Agricultural  Experiment  Station 


BULLETIN  No.  270 


MEASURING  THE  BREEDING  VALUE  OF  DAIRY 

SIRES  BY  THE  RECORDS  OF  THEIR  FIRST 

FEW  ADVANCED  REGISTRY  DAUGHTERS 

BY  F.  A.  DAVIDSON 


URBANA,  ILLINOIS,  JUNE,  1925 


CONTENTS 

PAGE 

INTRODUCTION 545 

SOURCE  OF  DATA 546 

OUTLINE  OF  INVESTIGATION 546 

Number  of  Tested  Daughters  Necessary  to  Measure  Relative  Breeding 

Value  of   Sire 548 

Smallest  Number  of  Tested  Daughters  Whose  Average  Production 

Will  Approximate  Average  of  Large  Number 548 

DISCUSSION  OF  RESULTS 556 

DERIVATION  OF  METHOD 560 

APPLICATION   OF  METHOD 564 

Cautions  in  Use  of  Method 564 

CONCLUSIONS 564 

LITERATURE    CITED..  .  565 


MEASURING  THE  BREEDING  VALUE  OF  DAIRY 

SIRES  BY  THE  RECORDS  OF  THEIR  FIRST 

FEW  ADVANCED  REGISTRY  DAUGHTERS 

By  F.  A.  DAVIDSON,  First  Assistant  in  Dairy  Husbandry 

INTRODUCTION 

The  breeding  value  of  dairy  sires  lies  mainly  in  their  ability  to 
transmit  to  their  offspring  factors  for  high  milk  and  butterfat  produc- 
tion. The  fact  that  all  sires  are  not  equal  in  this  respect  makes  it 
necessary  to  select  for  breeding  purposes  only  those  sires  transmitting 
the  largest  number  of  such  factors.  An  expression  of  this  transmitting 
ability  may  be  found  in  part  in  the  milk  and  butterfat  productions  of 
the  daughters,  and  it  is  by  means  of  these  productions  that  breeders 
in  general  have  measured  the  breeding  value  of  dairy  sires.  This 
method  of  measuring  breeding  values,  altho  used  extensively  at  the 
present  time,  is  open  to  several  objections. 

The  productions  of  the  daughters  used  in  measuring  the  breeding 
value  of  dairy  sires  are  those  that  are  recorded  in  the  advanced  registers 
of  the  various  purebred  dairy  cattle  associations;  daughters  with 
records  in  these  registers  have  fulfilled  certain  production  requirements 
and  are  spoken  of  as  tested  daughters.  The  tested  daughters  of  a  sire, 
in  most  cases,  represent  only  his  best  daughters,  which  are  a  small 
percentage  of  all  his  daughters.  This  selection  is  due  in  part  to  the 
requirements  of  the  advanced  registers,  but  in  the  main  to  the  fact  that 
under  the  present  system  of  advanced  registry  testing  it  is  exceedingly 
unprofitable  to  test  daughters  other  than  those  showing  signs  of  high 
producing  ability.  The  productions  of  only  the  tested  daughters,  there- 
fore, do  not  provide  an  absolute  basis  upon  which  to  measure  the 
breeding  value  of  the  sires.  However,  since  the  tested  daughters  of  all 
sires  have  been  subjected  to  the  same  type  of  selection,  their  produc- 
tions provide  a  uniform  basis  and  may  be  used  as  a  means  of  measure- 
ment with  the  limitation  that  they  will  give  only  relative  results. 

The  primary  object  in  measuring  the  breeding  value  of  'dairy  sires 
is  to  determine  whether  or  not  they  are  increasing  the  productions  of 
their  daughters  over  that  of  their  daughters'  dams.  Inasmuch  as  there 
is  convincing  evidence  that  the  inherited  producing  ability  of  the 
daughters  is  influenced  as  much  by  their  dams  as  by  their  sires,  the 
productions  of  the  daughters'  dams  as  well  as  of  the  daughters  should 
be  taken  into  account.  Since,  however,  in  many  dairy  herds  the  produc- 
tions of  very  few  of  the  dams  have  been  recorded,  the  recorded 

545 


546  BULLETIN  No.  270  [June, 

productions  of  the  daughters  provide  the  only  available  means  of 
measuring  the  sire's  breeding  value.  Altho  the  productions  of  the 
daughters  cannot  be  used  alone  to  measure  the  actual  breeding  value 
of  the  sire,  they  can  be  used  to  determine  the  approximate  breeding 
value. 

Breeders  in  general  have  found  that  the  average  of  the  milk  and 
butterfat  productions  of  a  large  number  of  tested  daughters  may  be 
used  as  a  relative  measure  of  the  transmitting  ability  of  the  sire.  The 
testing  of  a  large  number  of  daughters,  however,  necessitates  the  ex- 
penditure of  a  great  deal  of  time  and  money,  and  would  be  impractical 
in  the  majority  of  dairy  herds  at  the  present  time.  Furthermore,  very 
few  sires  are  kept  in  herds  long  enough  for  many  of  their  daughters  to 
be  tested,  being  sold  or  killed  long  before  their  breeding  value  is 
determined.  Many  good  sires  have  been  lost  in  this  way  and  many 
more  will  be  lost,  unless  some  means  is  provided  whereby  their  breed- 
ing value  can  be  determined  when  their  first  few  daughters  are  tested. 
W hat  is  needed  is  a  method  which  will  enable  breeders  to  predict,  with- 
in given  limits,  the  average  production  of  a  large  number  of  tested 
daughters  of  a  sire  from  the  average  production  of  his  first  few  tested 
daughters.  Such  a  method  will  save  breeders  a  great  deal  of  time  and 
money  and  will  also  be  an  important  factor  in  improving  the  trans- 
mitting ability  of  the  dairy  sire  population  in  general. 

In  view  of  these  facts,  an  investigation  was  planned  in  which  the 
milk  and  butterfat  productions  of  the  tested  daughters  of  a  large 
number  of  dairy  sires  were  studied.  The  purpose  of  the  investigation 
was  to  derive  a  method  by  which  the  productions  of  the  first  few  tested 
daughters  of  a  sire  might  be  used  as  a  criterion  to  measure  his  relative 
breeding  value. 

SOURCE  OF  DATA 

The  Register  of  Merit  of  the  American  Jersey  Cattle  Club  lists  the 
names  of  Jersey  sires  having  three  or  more  tested  daughters.  Under 
each  sire's  name  is  listed  the  names  of  his  tested  daughters.  Volumes 
1911  to  1921  inclusive  of  the  Register  of  Merit  contain  133  sires  having 
fifteen  or  more  tested  daughters  with  yearly  or  long-time  records.  The 
Jersey  Register  of  Merit  was  chosen  as  the  source  of  data  because  it 
contained  the  largest  number  of  sires  having  fifteen  or  more  tested 
daughters  with  yearly  records. 

OUTLINE  OF  INVESTIGATION 

The  investigation  as  planned  involved  three  steps: 

1.  The  determination  of  a  definite  large  number  of  tested  daughters 

of  a  sire,  the  average  of  whose  productions  could  be  used  as  a  relative 

measure  of  the  sire's  breeding  value. 


1925]  MEASURING  THE  BREEDING  VALUE  OF  DAIRY  SIRES  547 

2.  The  determination  of  the  smallest  number  of  first  tested  daugh- 
ters of  a  sire,  the  average  of  whose  productions  will  closely  approximate 
the   average    production   of   this   determined   large   number   of   tested 
daughters. 

3.  The  derivation  of  a  method  by  which  the  average  production  of 
the  determined  small  number  of  tested  daughters  of  a  sire  may  be  used 
to  predict  within  given  limits  the  average  production  of  the  determined 
large  number  of  tested  daughters. 

Before  proceeding  with  the  three  steps  outlined  above,  it  was  neces- 
sary to  correct  the  recorded  production  of  each  daughter  for  the  in- 
fluence of  age  of  cow  and  percentage  of  fat  in  the  milk.  The  correction 
for  the  influence  of  age  was  necessary  because,  on  the  average,  the 
production  of  a  cow  increases  up  to  the  age  of  eight  or  nine  years  and 
then  gradually  decreases.  This  influence  of  age  on  production  has  been 
demonstrated  very  clearly  by  Gowen  (1919),  Pearl  (1919),  and  Brody, 
Ragsdale,  and  Turner  (1923).  The  correction  for  age  was  accomplished 
by  reducing  the  production  record  (milk  and  fat)  of  each  daughter  to 
the  production  at  the  age  of  maximum  production,  which  for  Jerseys 
is  eight  years  and  one  month.  The  factors  used  for  these  corrections 
are  given  in  Table  6.  Owing  to  the  fact  that  the  percentage  of  fat  in  the 
milk  produced  changes  only  very  slightly  with  the  age  of  the  cow 
(Gowen  1919),  the  age-correction  factors  derived  for  the  milk  yield  can 
also  be  used  to  correct  the  fat  yield. 

The  productions  of  the  daughters  as  recorded  in  the  Register  of 
Merit  include  the  pounds  of  milk  and  the  pounds  of  butterfat  pro- 
duced, both  of  which  are  needed  to  secure  a  complete  expression  of 
each  daughter's  production.1  Therefore,  it  was  necessary  to  find  a 
single  expression  for  both  products.  Such  an  expression  may  be  secured 
by  reducing  the  milk  and  butterfat  productions  of  all  the  daughters  to 
milk  with  a  common  percentage  of  fat.  If  all  the  milk  productions  of 
the  daughters  contain  the  same  percentage  of  fat,  the  amount  of  fat  in 
the  milk  will  vary  directly  with  the  milk  yield  and  need  not  be  con- 
sidered separately.  By  using  the  formula  .4M-J-15F,  where  M  = 
recorded  milk  yield  in  pounds  and  F  —  recorded  fat  yield  in  pounds, 
all  milk  with  varying  fat  percentages  may  be  reduced  to  milk  contain- 
ing 4  percent  of  fat  (indicated  as  F.  C.  M.,  fat-corrected  milk).2 

In  the  statistical  study  which  follows,  the  recorded  productions  of 
all  the  daughters  have  been  corrected,  as  explained  above,  both  for 
influence  of  age  of  cow  and  for  percentage  of  fat  in  the  milk;  that  is, 
all  productions  are  reported  in  terms  of  A.-F.  C.  M.  (age-  and  fat- 
corrected  milk). 


contains  solid  material  other  than  fat,  collectively  known  as  solids-not-fat. 
It  requires  work  on  the  part  of  the  cow  to  produce  these  solids;  hence  they  should  be 
taken  into  consideration  as  well  as  the  fat  when  measuring  her  producing  ability. 

2This  formula  was  determined  by  Gaines  and  Davidson  (1923)  and  is  based  on 
the  energy  value  of  milk  at  varying  fat  percentages.  Bulletin  245  of  this  Experiment 
Station  gives  a  complete  discussion  of  the  derivation  of  this  formula. 


548 


BULLETIN  No.  270 


[June, 


NUMBER  OF  TESTED  DAUGHTERS  NECESSARY  TO  MEASURE  RELATIVE 
BREEDING  VALUE  OF  SIRE 

From  a  preliminary  statistical  study  of  the  production  records  of 
the  tested  daughters  of  dairy  sires  it  was  found  that  the  mathematical 
constants  measuring  the  variability  in  the  productions  of  the  first 
fifteen  tested  daughters  were  in  every  case  many  times  their  probable 
errors  and  were  changed  only  very  slightly  by  the  productions  of  addi- 
tional daughters.  The  mathematical  constants  measuring  the  variability 
in  the  productions  of  the  first  fifteen  tested  daughters  of  ten  of  the  133 
Jersey  sires  chosen  at  random  are  reported  in  Table  A. 

TABLE  A. — MEASURES  OF  VARIABILITY  IN  THE  A.-F.C.M.  PRODUCTIONS  OF  THE  FIRST 

FIFTEEN  TESTED  DAUGHTERS  OF  TEN  JERSEY  SIRES  CHOSEN  AT  RANDOM  FROM 

THE  133  JERSEY  SIRES 


Name  of  sire 

Mean 
(pounds  of 
A.-F.C.M.) 

Standard 
deviation 
(pounds  of 
A.-F.C.M.) 

Coefficient  of 
variability 
(percentage) 

Sayda's  Heir  3d  

11483  ±  347 

1990  ±  245 

17.3  =t  2.2 

Gamboge's  Knight  

11417  ±  271 

1556  ±  192 

13  6  =«=  1.7 

Hood  Farm  Torono  

12183  ±  375 

2152  ±  265 

17.7  ±  2.2 

Loretta's  King  

10250  ±  252 

1449  ±  178 

14.1  =»=  1.8 

Pogis  99th  of  Hood  Farm  

14383  ±  583 

3349  ±  412 

23  3  =»=  3.0 

Raleigh's  Fairy  Boy       .        

10783  ±  245 

1408  ±  173 

13  1  ±  1.6 

Royal  Majesty  of  St.  Cloud  

10917  ±  374 

2150  ±  265 

19.7  *  2.5 

Hood  Farm  Pogis  9th  

10450  ±  269 

1547  ±  191 

14.8  ±  1.9 

Imp.  Oxford  You'll  Do  

10417  ±  358 

2055  ±  253 

19.7  =*=  2.5 

Irene's  King  Pogis  

9983  ±  252 

1448  ±  178 

14.5  ±  1.8 

These  constants  in  Table  A  are  also  many  times  their  probable 
errors.  Hence  it  may  be  assumed  that  the  variability  in  the  productions 
of  the  first  fifteen  tested  daughters  of  a  Jersey  sire  is  representative  of 
the  variability  in  the  productions  of  any  larger  number  of  his  tested 
daughters.  Accordingly,  the  average  production  of  the  first  fifteen  tested 
daughters  of  a  Jersey  sire  may  be  used  as  a  relative  measure  of  his 
breeding  value  and  also  as  a  basis  with  which  may  be  compared  the 
average  production  of  any  smaller  number  of  his  tested  daughters. 

SMALLEST  NUMBER  OF  TESTED  DAUGHTERS  WHOSE  AVERAGE  PRODUCTION 
WILL  APPROXIMATE  AVERAGE  OF  LARGE  NUMBER 

In  order  to  determine  the  smallest  number  of  a  sire's  first  tested 
daughters  the  average  of  whose  productions  would  closely  approximate 
the  average  production  of  his  first  fifteen  tested  daughters,  it  was 
necessary  to  classify  the  tested  daughters  of  each  of  the  133  sires  in 
the  following  manner: 

1.  Each  daughter  of  each  sire  was  first  classified  chronologically, 
i.e.,  as  the  first,  second,  third,  etc.,  daughter  of  the  sire  to  appear  in  the 
Register  of  Merit. 


1925~\  MEASURING  THE  BREEDING  VALUE  OF  DAIRY  SIRES  549 

2.  According  to  the  first  classification,  the  daughters  of  each  sire 
were   then   separated   into   fifteen   daughter-groups.    These   daughter- 
groups    were    produced   by   cumulation    of    the    daughters    from    each 
preceding  group;  i.e.,  the  first  daughter  appearing  in  the  Register  of 
Merit  formed  the  first  group,  the  first  and  second  daughters  appearing 
in  the  Register  formed  the  second  group,  the  first,  second,  and  third 
daughters,    the    third    group,    etc.,    until    all    fifteen    daughters    were 
formed  into  the  fifteenth  group.   A  tabular  illustration  of  this  classifica- 
tion may  be  found  in  Table  B. 

3.  A  further  classification  was  then  made  by  combining  all  the 
daughter-groups   of   the    133    sires   into   fifteen    aggregates.    The   first 
aggregate  composed  all  the  first  daughter-groups,  the  second  aggregate 
composed   all   the  second  daughter-groups,   etc.,   the   fifteenth    aggre- 
gate composing  all  of  the  fifteenth  daughter-groups.   This  classification 
is  also  illustrated  in  Table  B. 

The  further  analysis  of  the  tested  daughters  of  the  133  sires  con- 
sisted of  comparisons  between  the  A.-F.  C.  M.  productions  of  the  first 
few  daughters  of  each  sire  and  the  A.-F.  C.  M.  productions  of  the  first 
fifteen  daughters  of  the  sire.  These  comparisons  were  made  as  follows: 

1.  The  average  A.-F.  C.  M.  production  or  mean  A.-F.  C.  M.  milk 
yield  of  each  daughter-group  was  determined,  i.e.,  the  productions  or 
milk  yields  of  the  daughters  in  each  group  were  added  and  the  sum 
divided  by  the  number  of  daughters  in  the  group.1    The  mean  milk 
yields  of  the  daughter-groups  in  each  aggregate  were  correlated  with  the 
mean  milk  yields  of  their  respective  fifteenth  daughter-groups  in  the 
fifteenth  aggregate.    In  other  words,  the  mean  milk  yields  of  all  the 
first  daughter-groups  were  correlated  with  the  mean  milk  yields  of  their 
respective  fifteenth  daughter-groups,  the  mean  milk  yields  of  all  the 
second  daughter-groups  were  correlated  with  the  mean  milk  yields  of 
their  respective  fifteenth  daughter-groups,  etc.,  until  every  daughter- 
group  was  correlated  with  its  respective  fifteenth  daughter-group.  The 
means  of  the  fifteenth  daughter-groups  were  in  every  case  used  as  the 
basis  with  which  were  correlated  the   means  of  the  other  daughter- 
groups.     By  making  this  comparison,   a  series  of  fifteen  correlation 
coefficients  was  set  up  which  indicated  the  relation  existing  between  the 
mean  milk  yields  of  the  daughter-groups   in  each  aggregate  and  the 
mean  milk  yields  of  their  respective  daughter-groups  in  the  fifteenth 
aggregate. 

2.  The   standard  deviation  of  the   milk  yields   of  the  individual 
daughters    in    each    aggregate    (Table    B)    was    compared    with    the 
standard  deviation  of  the  milk  yields  of  the  individual  daughters  in  the 
fifteenth  aggregate.    A  similar  comparison  was  made  between  the  co- 

1Hereinafter  for  the  sake  of  convenience  the  term  "milk  yield"  will  be  used 
interchangeably  with  the  term  "production."  It  must  always  be  remembered,  how- 
ever, that  "milk  yield"  refers  here  to  age-  and  fat-corrected  milk  and  not  to  the 
recorded  milk  yield. 


550 


BULLETIN  No.  270 


\_June, 


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1925]  MEASURING  THE  BREEDING  VALUE  OF  DAIRY  SIRES  551 

efficients  of  variability  of  the  milk  yields  of  the  individual  daughters 
in  the  aggregates.  This  comparison  was  necessary  in  order  to  determine 
whether  or  not  the  productions  of  the  first  few  tested  daughters  of  the 
133  sires  were  more  variable  than  the  productions  of  the  first  fifteen 
tested  daughters. 

3.  The  mean  milk  yield  of  the  fifteenth  daughter-group  of  each 
sire  was  subtracted  from  the  mean  milk  yield  of  each  smaller  group  of 
tested  daughters.  The  daughter-groups  of  the  133  sires  were  combined 
into  aggregates;  and  the  mean  milk  yields  of  the  daughter-groups 
in  the  fifteenth  aggregate  were  subtracted  from  the  mean  milk  yields 
of  their  respective  daughter-groups  in  each  of  the  other  aggregates.  In 
this  way  a  class  of  133  differences  was  set  up  for  each  aggregate,  there 
being  133  daughter-groups  in  each  aggregate.  The  series  of  differences 
for  each  aggregate  was  given  the  same  number  as  the  aggregate  from 
which  it  was  determined,  i.e.,  the  differences  between  the  mean  milk 
yields  of  the  daughter-groups  in  the  fifteenth  aggregate  and  the  mean 
milk  yields  of  their  respective  daughter-groups  in  the  first  aggregate 
formed  the  first  class,  etc.  These  classes  of  differences  are  shown 
graphically  in  Figs.  1  to  14.  The  mean  and  the  standard  deviation  of 
each  class  of  differences  were  determined.  A  comparison  was  then  made 
between  the  means  of  the  classes  of  differences  and  likewise  between 
the  standard  deviations. 

The  mean  of  the  first  class  of  differences  is  equivalent  to  the 
difference  between  the  mean  milk  yield  of  all  the  individual  daughters 
in  the  fifteenth  aggregate  and  the  mean  milk  yield  of  all  the  individual 
daughters  in  the  first  aggregate.  The  same  relation  holds  true  for  the 
mean  of  each  class  of  differences.  Hence  the  comparison  between  the 
means  of  the  classes  of  differences  indicates  the  relation  existing  be- 
tween the  mean  milk  yield  of  all  the  daughters  in  the  fifteenth  aggre- 
gate and  the  mean  milk  yield  of  all  the  daughters  in  each  of  the  other 
aggregates. 

The  standard  deviations  of  each  class  of  differences  measures  the 
variability  among  the  differences  in  each  class.  Hence  a  comparison 
between  these  standard  deviations  indicates  the  relation  existing 
between  classes  in  regard  to  the  variability  among  the  differences  with- 
in them.  In  other  words,  this  comparison  indicates  the  extent  to  which 
the  mean  milk  yields  of  the  daughter-groups  in  each  aggregate  deviated 
from  the  mean  milk  yields  of  their  respective  daughter-groups  in  the 
fifteenth  aggregate.  It  also  provides  another  means  of  expressing  the 
same  relation  as  indicated  by  the  correlations  in  the  first  comparison, 
i.e.,  it  determines  on  the  average  how  closely  the  mean  milk  yield  of 
each  daughter-group  approximates  the  mean  milk  yield  of  the  fifteenth 
daughter-group. 


552 


BULLETIN  No.  270 


[June, 


19251 


MEASURING  THE  BREEDING  VALUE  OF  DAIRY  SIRES 


553 


-33    -29    -25    -21    -17    -13     -9     -5     -l+l     +  5     +9    -H3    +  17    +n    -1-25    +zg   +33 


6  3  T   51  A  «•  HOa  11   97563211 


-33    -29    -25    -«    ~1T    -13     -9    -5     -  l-t-l    -1-5     +9    +13 


+17    +21    +«5   +29    +33 


*  Curss  Plio-PoiNTS 


-33    -£9    -25    -21    -IT    -13     -9     -5     -l+l     +5     +9    +13    +17    +«    +«5    +29    +33 


-   CLASS  f1iD-ft>MT<S 


-33    -25    -25    -21    -17    -13     -9     -5     -1+1     +5     +9 


554 


BULLETIN  No.  270 


[June, 


-33    -29    -25    -21    -17    -13    -9    -5     -1+1     +5     +9    +13    +17 

MlLK-CwT-   Class  ttlD-POINTS 


+25    +2?    +33 


-33    -29   -25   -21    -17-13     -9    -5    -l+l    +5    +9   +13    +17    +a  +25  +29    +33 


-33    -29    -25    -21    -17    -13     -9     -5     -1+1     +5     +9    +13    +U   +»1    +*5    +«9   +33 


I. 


CIQ.  I O -TENTH 

OF  DIFFERENCED 


-33    -29    -25    -21    -17    -13    -9    -5     -i+i     +5     +9    +13    +17    +21    +25    +29    +33 


1925] 


MEASURING  THE  BREEDING  VALUE  OF  DAIRY  SIRES 


555 


JO 


Fio.  //-  ELEVENTH 

Of  DIFFERENCES 


-»  -2?   -25   -a   -17  -13    -9    -5    -1-41    +  5    +9  413  -HT  fa  415   ^25  433 

T'  CLASS  tlio  POINTS 


-33    -29    -25    -n    -17    -13    -9     -5    -i-H     +5     ^9    +*3    +17 


FIG. /<5 -THIRTEENTH  Cut 

OF  D/FFE&EMCE3 


-33    -29    -25    -21    -l 


-13     -9     -5     -141     +  5     49    413    +17    +21    +25    429    4J3 
MlLK-CwT*  CLMsHlO-PoiHTO 


-33    -29    -25    -21    -»7    -»3    -9    -5     -1-fi     45     +9    413    417    4«    425    +V)    433 


556 


BULLETIN  No.  270 


[June, 


DISCUSSION  OF  RESULTS 

The  coefficients  of  the  correlations  between  the  mean  milk  yields 
of  the  daughter-groups  in  each  aggregate  and  the  mean  milk  yields  of 
their  respective  daughter  groups  in  the  fifteenth  aggregate,  are  reported 
in  Table  1  and  shown  graphically  in  Fig.  A.  The  fitted  curve  in  Fig.  A, 

TABLE  1. — COEFFICIENTS  OF  THE  CORRELATIONS  BETWEEN  THE  MEAN  MILK.  YIELDS  OF 

THE  DAUGHTER-GROUPS  IN  EACH  AGGREGATE  AND  THE  MEAN  MILK  YIELDS  OF 

THE  RESPECTIVE  DAUGHTER-GROUPS  IN  THE  FIFTEENTH  AGGREGATE 


Aggregates 

Coefficients  of  Correlations 

Observed 

Calculated 

1         

.552  ±  .041 
.777  ±  .023 
.804±  .027 
.895  ±  .012 
.898  ±  .011 
.910  ±  .010 
.926  ±  .008 
.930  =«=  .008 
.947  ±  .006 
.959  ±  .005 
.965  ±  .004 
.976  ±  .003 
.981  ±  .002 
.987  ±  .002 
1.000±  .000 

.526 
.737 
.823 
.869 
.899 
.919 
.935 
.947 
.957 
.965 
.973 
.981 
.988 
.994 
1.000 

2  

3  

4    

5       

6       

7  

8  

9  

10  

11  

12  

13  

14  

15  

representing  the  relation  between  these  coefficients,  rises  very  rapidly 
at  first  and  then  gradually  flattens  out  into  an  almost  straight  line.  The 
point  at  which  this  flattening  in  the  curve  begins  to  be  pronounced  lies 
in  the  region  of  the  6  coefficient  of  correlation,  which  represents  the 


5          <          7          «          9          10        11        U        13        1»        15 


1925~]  MEASURING  THE  BREEDING  VALUE  OF  DAIRY  SIRES  557 

sixth  daughter-groups.  The  equation  to  this  curve  is  given  in  Table  7. 
In  Table  1  the  coefficients  calculated  from  the  fitted  curve  increase 
from +.526,  the  coefficient  of  the  first  aggregate,  to +.919,  the  co- 
efficient of  the  sixth  aggregate,  and  from  +.919,  the  coefficient  of  the 
sixth  aggregate,  to  +  1.000,  the  coefficient  of  the  fifteenth  aggregate. 
Approximately  three-quarters  of  the  total  increase  in  the  correlations 
from  the  first  to  the  fifteenth  aggregate  is  reached  by  the  sixth  aggre- 
gate, thus  showing  that  after  the  sixth  aggregate  additional  daughters 
influenced  the  correlations  only  very  slightly.  The  coefficient  of  +  .919 
for  the  sixth  aggregate  also  represents  a  very  significant  correlation  and, 
together  with  the  above  relation  described  by  the  fitted  curve,  indicates 
that  on  the  average  the  mean  milk  yields  of  the  first  six  tested  daughters 
of  the  133  sires  are  a  very  close  approximation  to  the  mean  milk  yields 
of  their  first  fifteen  tested  daughters. 

The  standard  deviations  and  coefficients  of  variability  of  the  milk 
yields  of  the  individual  daughters  in  each  aggregate  are  reported  in 
Table  2  and  shown  graphically  in  Fig.  B.  The  fitted  curves  in  Fig.  B, 
representing  the  relation  between  the  standard  deviations  and  the 
relation  between  the  coefficients  of  variability,  are  straight  lines  which 
rise  only  very  slightly  from  the  first  to  the  fifteenth  aggregate.  The 
equations  to  these  curves  are  given  in  Table  7.  The  relations  described 
by  these  fitted  curves  in  Fig.  B  indicate  that  on  the  average  the  vari- 
ability in  the  milk  yields  of  the  first  six  tested  daughters  of  the  133 
sires  is  practically  the  same  as  the  variability  in  the  milk  yields  of  the 
first  fifteen  tested  daughters.  In  fact,  the  variability  in  the  milk  yields 
of  the  tested  daughters  in  any  aggregate  differs  only  very  slightly  from 
the  variability  in  any  other  aggregate.  Hence  it  may  safely  be  assumed 
that  the  results  from  the  first  and  third  comparisons  are  not  due  to 
chance  differences  in  the  variability  within  the  milk  yields  of  the  tested 
daughters  in  the  aggregates. 

The  mean  and  the  standard  deviation  of  each  class  of  differences 
are  reported  in  Table  3  and  shown  graphically  in  Fig.  C.  The  fitted 
curve  in  Fig.  C,  representing  the  relation  between  the  standard  devia- 
tions, falls  very  rapidly  at  first  and  then  gradually  flattens  out  into  an 
almost  straight  line.  Here  again  the  point  at  which  this  flattening  in 
the  curve  begins  to  be  pronounced  is  in  the  region  of  the  sixth  standard 
deviation,  which  represent  the  sixth  daughter-groups.  The  equation  to 
this  curve  (Table  7)  is  of  the  same  type  as  the  equation  to  the  curve 
expressing  the  relation  between  the  coefficients  of  correlation  in  Fig.  A. 
In  Table  3  the  means  calculated  from  the  fitted  curve  are  very  small 
and  can  hardly  be  said  to  indicate  any  general  relation. 

In  the  same  table  the  standard  deviations  calculated  from  the  fitted 
curve  decrease  from  1,829  pounds,  the  standard  deviation  of  the  first 
class  of  differences,  to  602  pounds,  the  standard  deviation  of  the  sixth 
class  of  differences,  and  from  602  pounds,  the  standard  deviation  of  the 
sixth  class  of  differences,  to  155  pounds,  the  standard  deviation  of  the 


558 


BULLETIN  No.  270 


[June, 


TABLE  2. — MEASURES  OF  VARIABILITY  IN  THE  MILK  YIELDS  or  THE 
INDIVIDUAL  DAUGHTERS  IN  EACH  AGGREGATE 


Aggre- 
gates 

Number 
of 
daughters 

Mean 
(pounds  of 
A.-F.C.M.) 

Standard  deviation 
(pounds  of  A.-F.C.M.) 

Coefficient  of  variability 
(percentage) 

Observed 

Calculated 

Observed 

Calculated 

1  .. 

133 

11754  ±  138 

2277  ±94.2 

2314 

19.4  ±  .83 

20.22 

2.  .. 

266 

11669  ±    94 

2264  ±66.2 

2317 

19.4  ±  .59 

20.23 

3.  .. 

399 

11578  ±    78 

2312  ±55.2 

2320 

20.0  ±  .50 

20.23 

4.  .. 

532 

11593  ±    70 

2402  ±49.7 

2323 

20.7  ±  .45 

20.24 

5... 

665 

11515  ±    63 

2392  ±44.2 

2325 

20.8  ±  .40 

20.24 

6.  .. 

798 

11498  ±    57 

2363  ±39.9 

2328 

20.6  ±  .36 

20.25 

7  ... 

931 

11478  ±    51 

2324  ±36.3 

2331 

20.3  ±  .33 

20.25 

8... 

1064 

11472  ±    48 

2319  ±33.9 

2334 

20.2  ±  .31 

20.26 

9... 

1197 

11463  ±    45 

2312  ±  31.9 

2336 

20.2  ±  .29 

20.26 

10... 

1330 

11469  ±    43 

2319  ±30.3 

2339 

20.2  ±  .27 

20.27 

11.  .. 

1463 

11464  ±    41 

2315  ±28.9 

2342 

20.2  ±  .26 

20.27 

12... 

1596 

11482±    39 

2323  ±27.7 

2345 

20.2  ±  .25 

20.28 

13... 

1729 

1  1499  ±    38 

2364  ±27.1 

2347 

20.  6  ±  .25 

20.28 

14... 

1862 

11475  ±    37 

2355  ±26.0 

2350 

20.5  ±  .24 

20.29 

15... 

1995 

11453  ±    36 

2362  ±25.2 

2353 

20.6  ±  .23 

20.29 

fourteenth  class  of  differences.  The  relation  between  these  standard 
deviations  indicates  that  the  variability  in  the  differences  between  the 
mean  milk  yields  of  the  daughter-groups  in  the  fifteenth  aggregate  and 
the  mean  milk  yields  of  their  respective  daughter-groups  in  each  of  the 
other  aggregates  decreases  markedly  from  the  first  to  the  sixth  aggre- 
gate and  from  there  on  decreases  only  very  slightly.  In  other  words, 
the  first  six  tested  daughters  of  the  133  sires  is  the  smallest  number  of 
tested  daughters  whose  mean  milk  yields  do  not  deviate  very  widely 
from  the  mean  milk  yields  of  the  first  fifteen  tested  daughters.  This 

TABLE  3. — MEASURES  OF  VARIABILITY  IN  EACH  CLASS  OF  DIFFERENCES1 


Class  of  differences 

Mean 
(pounds  of  A.-F.C.M.) 

Standard  deviation 
(pounds  of  A.-F.C.M.) 

Observed 

Calculated 

Observed 

Calculated 

1  .  . 

+304 
+219 
+  121 
+  144 
+  53 
+  38 
+  13 
+  13 
+     7 
+  13 
+     4 
+  23 
+  47 
+  28 

+321 
+  193 
+  129 
+  91 
+  66 
+  49 
+  36 
+  27 
+  21 
+  16 
+  14 
+  13 
+  13 
+  15 

1903  ±  78.7 
1070  ±44.  2 
970  ±  40.1 
823  ±  34.0 
707  ±29.3 
616  ±25.5 
566  ±  23.0 
510  ±21.1 
471  ±  19.5 
372  ±  15.4 
342  ±  14.1 
285  ±  11.8 
196  ±    8.1 
128  ±    5.3 

1829 
1252 
969 
801 
687 
602 
534 
476 
423 
372 
322 
269 
214 
155 

2  

3  

4  

5  

6  

7  

8  

9  

10  

11  

12  

13  

14  

*Each  class  of  differences  was  determined  by  subtracting  the  mean  milk  yields  of  the 
daughter-groups  in  the  fifteenth  aggregate  from  the  mean  milk  yields  of  the  respective 
daughter-groups  in  each  of  the  other  aggregates. 


19251 


MEASURING  THE  BREEDING  VALUE  OF  DAIRY  SIRES 


559 


relation  is  also  illustrated  very  clearly  by  comparing  the  polygons  in 
Figs.  1  to  14. 


stoo 

o        o 

O         Ul 

—0  o— 

* 

* 

x 

M 

X            K 

16$ 

I 

Ul 

Q 

O      £00 

/*?.  B 
n        n    STANDARD    DEVIAJIONG 

.* 

r" 

rrs  Of  VM 

LABILITY 

3       *       $        6        7        8       9        10      u      12      13      1>» 
AGGREGATES 


The  mathematical  constants  in  Tables  1,  2,  and  3,  which  describe 
the  relation  existing  between  the  milk  yields  of  the  first  fifteen  tested 
daughters  of  the  133  Jersey  sires  taken  as  a  whole,  are  all  many  times 


their  probable  errors.  Hence  it  may  be  assumed  that  these  tested 
daughters  of  the  133  Jersey  sires,  as  classified  into  the  daughter-groups, 
aggregates,  etc.,  are  representative  of  the  corresponding  tested 
daughters  of  all  Jersey  sires.  Following  this  supposition  it  may  also  be 
assumed  that  on  the  average  the  above  mathematical  constants  repre- 


560 


BULLETIN  No.  270 


[June, 


sent  the  relations  existing  between  the  milk  yields  of  the  first  fifteen 
tested  daughters  of  any  Jersey  sire.  Accordingly  then,  a  method  can  be 
derived  by  which  the  mean  milk  yield  of  the  first  fifteen  tested 
daughters  of  any  Jersey  sire  can  be  predicted  to  fall  within  given  limits 
from  the  mean  milk  yield  of  any  smaller  number  of  his  first  tested 
daughters. 

DERIVATION  OF  METHOD 

Each  class  of  differences,  as  shown  graphically  in  Figs.  1  to  14, 
tends  to  approximate  a  normal  frequency  curve,  some  of  the  curves, 
however,  being  distinctly  of  the  "cocked  hat"  type.  The  mean  of  any 
class  of  differences  ±  2.141  times  the  standard  deviation  of  that  class 
gives  limits  such  that  the  odds  are  30  to  1  that  any  single  difference, 

TABLE  4. — LIMITSX  AT  ODDS  OF  30  TO  1  AND  100  TO  1  FOR  EACH  GROUP  OF 
FIRST  TESTED  DAUGHTERS  OF  ANY  JERSEY  SIRE 


Daughter  groups 

Limits  at  odds  of  30  to  1 
(pounds  of  A.-F.C.M.) 

Limits  at  odds  of  100  to  1 
(pounds  of  A.-F.C.M.) 

1.  . 

+4235  to  -3593 

+5040  to  -4398 

2  

+2872  to  -2486 

+3423  to  -3037 

3  

+2203  to  -1945 

+2629  to  -2371 

4  

+  1805  to  -1623 

+2158  to  —1976 

5  

+  1536  to  -1404 

+  1838  to  -1706 

6  

+  1337  to  -1239 

+  1602  to  —1504 

7  

+  1179  to  -1107 

+  1414  to  -1342 

8  

+  1046  to  —  992 

+  1255  to  —1201 

9  

+  926  to  -  884 

+  1112  to  -1070 

10  

+  812  to  -  780 

+  976  to  -  944 

11  

+  703  to  -  675 

+  845  to  —  817 

12  ... 

+  589  to  —  563 

+  707  to  —  681 

13  

+  471  to  -  445 

+  565  to  -  539 

14  

+  347  to  -  317 

+  415  to  -  385 

limits  were  determined  from  the  calculated  means  and  standard  deviations  of 
the  classes  of  differences. 

determined  by  the  above  methods  for  that  class,  will  lie  within  them. 
Any  single  difference  in  a  class  of  differences  represents  the  difference 
between  the  mean  milk  yield  of  a  corresponding  daughter-group  and 
the  mean  milk  yield  of  the  fifteenth  daughter-group  of  an  individual 
sire.  Hence  the  mean  of  a  class  of  differences  ±2.14  times  the  standard 
deviation  of  that  class  gives  limits  such  that  the  chances  are  30  to  1  that 
the  difference  between  the  mean  milk  yields  of  the  corresponding  group 
of  tested  daughters  and  the  fifteenth  group  of  tested  daughters  of  any 
Jersey  sire,  will  lie  within  them.2  For  example,  the  mean  of  the  sixth 

The  constant  2.14  was  determined  by  means  of  the  equation  given  on  page 
XVIII  of  Karl  Pearson's  Tables  for  Statisticians. 

"Odds  of  30  to  1  are  considered  by  most  investigators  as  being  great  enough  to 
be  dependable;  however,  if  greater  odds  are  desired  2.58  X  a  will  give  odds  of  100  to  1. 
The  limits  for  these  odds  are  also  included  in  Table  5. 


1925} 


MEASURING  THE  BREEDING  VALUE  OF  DAIRY  SIRES 


561 


class  of  differences  ±  2.14  times  the  standard  deviation  of  that  class, 
gives  limits  of  -(-  1,337  pounds  to —  1,239  pounds.1  The  chances  are 
30  to  1  then  that  the  difference  between  the  mean  milk  yield  of  the  first 
fifteen  tested  daughters  and  the  mean  milk  yield  of  the  first  six  tested 
daughters  of  any  Jersey  sire  will  fall  within  them.2  In  other  words, 
the  chances  are  30  to  1  that  the  mean  milk  yield  of  the  first  six  tested 
daughters  of  any  Jersey  sire  will  not  be  more  than  1,337  pounds 
above  nor  less  than  1,239  pounds  below  the  mean  milk  yield  of  his 
first  fifteen  tested  daughters.  In  like  manner  similar  limits  were 
determined  for  each  group  of  tested  daughters.  These  limits  are  re- 
ported in  Table  4  and  shown  graphically  in  Fig.  D. 


+5000 


+  1(000 


-3000 


5       6       T       e       9 

DaUGHTER     G-&OUP3 

Hence  limits  may  be  found  (Table  4)  within  which  the  chances  are 
30  to  1  that  the  mean  milk  yield  of  any  small  group  of  a  sire's  first 
tested  daughters  will  deviate  from  the  mean  milk  yield  of  his  first 
fifteen  tested  daughters. 

lrThe  means  and  the  standard  deviations  used  to  determine  these  limits  were  the 
calculated  means  and  standard  deviations  given  in  Table  3. 

"Any  difference  between  the  mean  milk  yield  of  the  fifteenth  group  of  tested 
daughters  and  the  mean  milk  yield  of  any  smaller  group  of  tested  daughters  is  always 
determined  by  subtracting  the  mean  milk  yield  of  the  fifteenth  group  from  the  mean 
milk  yield  of  the  smaller  group. 


562 


BULLETIN  No.  270 


TABLE  5. — AGE-CORRECTION  FACTORS  FOR  GUERNSEY  MILK  YIELDS 
Age  refers  to  the  age  of  the  cow  at  date  of  last  calving  preceding  the  commencement  of 
the  record. 


(Additional  months) 


1  year 


2  years 


3  years 


4  years 


0 

1 

2 

3 

4 

5 

6 

7 1.61443 

8 1.56493 

9 1.54841 

10 .49109 

11 46191 

5  years 

0...  .08633 

1 .08214 

2 07810 

3 07419 

4 07041 

5 .06676 

6 06324 

7 .05983 

8 05655 

9 05338 

10 05032 

11 1.04737 

9  years 

0...  1.00010 

1 .00025 

2 .00047 

3 00076 

4 00113 

5 .00155 

6 .00205 

7 00263 

8 .00326 

9 .00397 

10 00475 

11 00559 

3  years 

0 .09107 

1 .09508 

2 .09920 

3 10342 

4 . 10776 

5 .11221 

6 .11678 

7 .12147 

8 .12628 

9 13121 

10 13628 

11 .14146 


1.43613 

1.41304 

1.39210 

1 . 37294 

1.35527 

1.33888 

1.32359 

.30926 

.29578 

.28306 

.27103 

. 25962 


.24875 
.23840 
. 22854 
.21910 
.21007 
.20142 
.19312 
.18514 
.  17748 
.17011 
. 16302 
.15619 


. 14961 
. 14326 
.13714 
.13123 
.12552 
.12001 
.11469 
.10955 
.10459 
.09979 
.09515 
1.09067 


6  years 


7  years 


8  years 


1.04453 
.04179 
.03916 
.03663 
.03420 
.03186 
.02961 
.02746 
.02538 
.02342 
.02154 

1.01974 


.01803 
.01641 
.01486 
.01339 
.01201 
.01071 
.00948 
.00834 
.00726 
.00627 
.00535 
.00451 


1.00374 
.00304 
.00242 
.00186 
.00139 
.00098 
.00064 
.00038 
.00018 
.00005 
.00000 
.00001 


10  years 


1  years 


2  years 


.00651 
.00750 
.00856 
.00969 
.01089 
.01216 
.01351 
.01493 
.01641 
.01798 
.01962 
.02133 


.02312 
.02499 
.02693 
.02894 
.03105 
.03322 
.03548 
.03782 
.04024 
.04274 
.04533 
.04800 


.05076 
.05361 
.05654 
.05956 
.06268 
.06589 
.06919 
.07259 
.07608 
.07968 
.08338 
.08717 


14  years 


15  years 


16  years 


1.14678 
1.15224 
.15784 
.16358 
.16946 
.17549 
.18167 
.18802 
. 19452 
.20118 
.20802 
.21502 


1.22220 
.22957 
.23712 
.24487 
.25282 
.26096 
.26932 
.27789 
.28669 
.29571 
.30497 
.31447 


EXPLANATION  OF  TABLE. — The  factors  given  in  this  table  were  derived  from  the 
equation:  Y  =  7017.4  +  328.0  X  -  12.4  X2  +  1729.2  Log,o  X.  (Y  =  yield  of  milk  in 
pounds;  X  =  age  in  units  of  six  months  commencing  at  1  year  and  3  months  for  the  zero 
point.)  This  equation  was  determined  by  the  author,  and  is  the  equation  of  the  curve  ex- 
pressing the  influence  of  age  on  the  milk  yield  of  Guernsey  cows.  It  is  based  upon  3,000 
yearly  milk  records  of  cows  as  published  in  Vols.  12  to  34  of  the  Guernsey  Advanced 
Register.  According  to  the  above  equation,  the  milk  yield  at  a  given  age  multiplied  by  the 
factor  given  in  the  table  for  that  age,  reduces  the  yield  to  that  of  the  age  of  maximum 
production,  that  is,  to  the  yield  at  8  years  and  10  months. 


1925] 


MEASURING  THE  BREEDING  VALUE  OF  DAIRY  SIRES 


563 


TABLE  6. — AGE-CORRECTION  FACTORS  FOR  JERSEY  MILK.  YIELDS 
Age  refers  to  the  age  of  the  cow  at  date  of  last  calving  preceding  the  commencement  of 
the  records. 


(Additional  months) 


1  year 


2  years 


3  years 


4  years 


0 2.19925 

1...  2.03645 

2...  .92160 

3...  83385 

4...  76336 

5  ..  .70474 

6...  .65481 

7...  61134 

.57282 
.53846 
.50761 
.47951 

5  years 

0...  1.07285 

1...          1.06849 

2 1.06440 

3...  1.06045 

4...       1.05664 

5...        1.05296 

6 1.04943 

7 1.04603 

1.04286 
1.03972 
1.03681 
1.03402 

9  years 

0...  .00543 

1 .00644 

2 00766 

3 00888 

4 .01020 

5 .01163 

6 01317 

7 .01471 

.01647 

9 01823 

10 1.02010 

11 1  02218 

13  years 

0...  .18120 

1 .18835 

2 19574 

3 20351 

4 .21153 

5 21966 

6 .22790 

7 23655 

8 24533 

9 25455 

10 26390 

11...  ...     .27355 


.45349 
.42959 
.40766 
.38735 
.36836 
.35044 
.33351 
.31752 
.30259 
.28833 
.27486 
.26215 


.25000 
.23839 
.22730 
.21669 
.20656 
. 19689 
.18779 
.17911 
.17069 
.16252 
1.15473 
1.14732 


6  years 


7  years 


1.14012 
1.13327 
1.12663 
1.12020 
1.11408 
1.10816 
1.10241 
1.09709 
1.09182 
1.08684 
1.08202 
1.07735 


8  years 


.03135 

.02870 

.02627 

.02396 

.02176 

1.01968 

1.01771 

1.01585 

1.01410 

1.01245 

1.01092 

1.00949 


1.00817 
1.00695 
1.00583 
1.00482 
1.00392 
1.00311 
1.00241 
1.00170 
1.00120 
1.00080 
1.00050 
1.00020 


.00010 
.00000 
.00010 
.00020 
.00040 
.00070 
.00110 
.00160 
.00220 
.00280 
.00361 
.00452 


10  years 


11  years 


12  years 


1.02428 
1.02648 
.02881 
.03125 
.03370 
.03627 
.03907 
.04188 
.04482 
.04789 
.05108 
.05441 


1.05775 
.06123 
.06485 
.06872 
.07262 
.07666 
.08073 
.08495 
.08944 
.09397 
.09866 
.10351 


1.10853 
.11359 
.11894 
. 12435 
. 12994 
.13572 
.14168 
. 14784 
.15407 
. 16050 
. 16727 
.17412 


14  years 


5  years 


16  years 


1.28351 
1.29375 
1.30430 
1.31517 
1.32637 
1.33791 
1 . 34982 
1.36208 
1.37471 
1.38773 
1.40118 
1.41507 


. 42939 
.44418 
.45946 
.47521 
.49151 
. 50840 
.52583 
. 54386 
. 56254 
.58187 
.60191 
.62267 


EXPLANATION  OF  TABLE. — The  factors  given  in  this  table  were  derived  from  the 
equation:  Y  =  4586.5  +  307.6  X  -  12.7  X2  +  2216.6  LogioX.  (Y  =  yield  of  milk  in 
pounds;  X  =  age  in  units  of  six  months  commencing  at  nine  months  for  the  zero  point.) 
This  equation  was  determined  by  John  W.  Gowen  (Bui.  281,  Maine  Agr.  Exp.  Sta.)  and 
is  the  equation  of  the  curve  expressing  the  influence  of  age  on  the  milk  yields  of  Jersey 
cows.  It  is  based  on  2,153  yearly  milk  records  of  cows  as  published  in  Vols.  1911  and  1913 
of  the  Jersey  Registry  of  Merit. 


564  BULLETIN  No.  270  [June, 

APPLICATION  OF  METHOD 

In  order  to  illustrate  the  accuracy  of  these  limits  and  also  their 
possible  application  to  the  tested  daughters  of  Guernsey  sires,  five 
Jersey  sires  not  included  in  the  original  133  sires,  and  five  Guernsey 
sires  with  fifteen  or  more  tested  daughters,  were  chosen  at  random, 
and  the  differences  between  the  mean  milk  yields  of  their  first  fifteen 
tested  daughters  and  the  mean  milk  yields  of  each  smaller  group  of 
first  tested  daughters  determined  by  the  above  procedure.  These  differ- 
ences are  given  in  Table  8.  The  limits  for  each  group  of  tested 
daughters,  such  that  the  chances  are  30  to  1  that  any  of  the  above  corres- 
ponding differences  should  fall  within  them,  are  also  given  in  Table  8. 

In  comparing  the  difference  for  each  group  of  daughters  of  the 
Jersey  sires  with  its  corresponding  limits,  it  will  be  found  that  in  every 
case  the  difference  falls  within  them.  These  limits,  therefore,  appear  to 
provide  a  dependable  method  by  which  the  average  productions  (mean 
milk  yields)  of  the  first  fifteen  tested  daughters  of  Jersey  sires  can  be 
predicted  from  the  average  productions  (mean  milk  yields)  of  any 
smaller  number  of  their  first  tested  daughters.  From  Table  8  it  would 
seem  that  these  limits  may  also  be  used  in  predicting  the  average  pro- 
ductions of  the  first  fifteen  tested  daughters  of  Guernsey  sires  from  the 
average  productions  of  any  smaller  number  of  their  first  tested  daughters. 

CAUTIONS  IN  USE  OF  METHOD 

In  using  the  limits  set  forth  in  Table  4,  it  must  always  be 
remembered  that  they  refer  to  the  corrected  milk  and  butterfat  produc- 
tions of  the  tested  daughters  and  not  to  their  recorded  productions.  The 
method  for  correcting  the  recorded  productions  of  the  tested  daughters 
of  Guernsey  and  Jersey  sires  is  described  on  page  547  and  in  Tables 
5  and  6  of  this  bulletin.  It  must  also  be  remembered  that  on  the  aver- 
age the  first  six  tested  daughters  of  a  sire  is  the  smallest  number  of 
tested  daughters  the  average  of  whose  productions  closely  approximates 
the  average  production  of  his  first  fifteen  tested  daughters.  The  limits 
for  smaller  groups  of  tested  daughters  are  much  wider  and  only  in  ex- 
ceptional cases  would  they  be  of  much  practical  value. 

CONCLUSIONS 

The  results  from  a  statistical  study  of  the  variability  within  the 
corrected  milk  and  butterfat  productions  of  the  tested  daughters  of  133 
Jersey  sires  are  as  follows:1 

1.  In  general  the  average  production  and  variability  among  the 
productions  of  the  first  fifteen  tested  daughters  of  a  Jersey  sire  are 
representative  of  the  average   production   and  variability   among  the 
productions  of  any  larger  number  of  the  sire's  tested  daughters. 

2.  On  the  average,  the  first  six  tested  daughters  of  a  Jersey  sire  is 
the  smallest  number  of  tested  daughters  the  average  of  whose  produc- 

The  recorded  milk  and  butterfat  productions  of  the  tested  daughters  were 
corrected  for  age  of  cow  and  percentage  of  fat  in  the  milk.  See  page  547  of  text. 


1925~\  MEASURING  THE  BREEDING  VALUE  OF  DAIRY  SIRES  565 

TABLE  7. — EQUATIONS  TO  FITTED  CURVES  IN  FIGS.  A,  B,  AND  C 
Fig.  A 

AQ7 

Coefficients  of  correlations:  Y  =  x+  3?9  +  1.0256  +  .000006X3 

Fig.  B 

Standard  deviations:  Y  =  2311.4  +  2.765X 

Coefficients  of  variability:  Y  =  20.219  +  .0051 IX 

Fig.C 

Standard  deviations:  y  =      3129      +  ^  ?  _   ^ 

A.  "r  .oo2 

Means:  Y=-™_- 63.6 +  .01X' 

-X.  +   1 

NOTE. — X  in  every  curve  is  a  variable  in  units  from  1  to  15,  and  Y  is  the  variable 
whose  quantity  is  determined  by  the  equation. 

tions  closely  approximates  the  average  production  of  the  first  fifteen 
tested  daughters  of  the  sire. 

3.  A  numerical  measure  of  this  closeness  of  agreement  between  the 
average  productions  of  the  first  six  tested  daughters  and  the  first  fifteen 
tested  daughters  of  a  Jersey  sire  is  as  follows: 

The  chances  are  30  to  1  that  the  average  production  of  the  first  six 
tested  daughters  of  a  Jersey  sire  will  not  be  more  than  1,337  pounds 
above  nor  less  than  1,239  pounds  below  the  average  production  of  the 
first  fifteen  tested  daughters.  In  other  words,  this  relation  between  the 
average  productions  of  the  first  six  tested  daughters  and  the  first  fifteen 
tested  daughters  of  a  Jersey  sire  can  be  expected  to  hold  true  for  30  out 
of  every  31  Jersey  sires. 

4.  Breeders  in  general  have  found  that  the  average  production  of 
a  large  number  of  tested  daughters  of  a  sire  can  be  used  as  a  relative 
measure  of  the  sire's  breeding  value.1    The  first  six  tested  daughters  of 
a  Jersey  sire  is  the  smallest  number  of  first  tested   daughters   whose 
average  production  can  be  used  as  a  means  to  measure  the  approximate 
breeding  value  of  the  sire. 

LITERATURE  CITED 

BRODY  S.,   RACSDALE,  A.  C.  and  TURNER,  C.  W.    Rate  of  growth  of  the  dairy  cow. 

Jour,  of  Gen.  Physiol.  6,  21-40.     1923. 
GAINES,  W.  L.  and  DAVIDSON,  F.  A.   Relation  between  percentage  fat  content  and  yield 

of  milk.   111.  Agr.  Exp.  Sta.  Bui.  245.    1923. 
GOWEN,  J.  W.    Animal  husbandry  investigations  in   1919.    Maine  Agr.  Exp.  Sta.  Ann. 

Rot.  1919,  249-284.    1919. 
PEARL,  R.  and  MINER,  J.  C.   Variation  of  the  milk  of  Ayrshire  cows  in  quantity  and 

fat  content  of  their  milk.   Maine  Agr.  Exp.  Sta.  Bui.  279.    1919. 
PEARSON,  KARL.  Tables  for  statisticians  and  biometricians.   Cambridge  University  Press. 

1914. 


1Owing  to  the  equal  influence  of  both  the  sire  and  dam  upon  the  inherited 
producing  ability  of  the  daughters,  and  also  to  the  fact  that  the  tested  daughters  of  a 
sire  in  most  cases  represent  only  the  best  daughters,  this  average  can  be  used  only  as 
a  relative  measure  of  the  breeding  value  of  the  sire.  See  page  545  of  text. 


566 


BULLETIN  No.  270 


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UNIVERSITY  OF  ILLINOIS-URBAN* 


